Some remarks on the geometry of circle maps with a break point

Abstract

We study circle homeomorphisms f ? C2(S1\{xb}) whose rotation number ?f is irrational, with a single break point xb at which f' has a jump discontinuity. We prove that the behavior of the ratios of the lengths of any two adjacent intervals of the dynamical partition depends on the size of break and on the continued fraction decomposition of ?f. We also prove a result analogous to Yoccoz?s lemma on the asymptotic behaviour of the lengths of the intervals of trajectories of the renormalization transformation Rn(f).